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Express in terms of sine and cosine Calculator

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1

Solved example of express in terms of sine and cosine

$\frac{1-\tan\left(x\right)}{1+\tan\left(x\right)}$
2

Applying the tangent identity: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$

$\frac{1+\frac{-\sin\left(x\right)}{\cos\left(x\right)}}{1+\tan\left(x\right)}$
3

Combine $1+\frac{-\sin\left(x\right)}{\cos\left(x\right)}$ in a single fraction

$\frac{\frac{-\sin\left(x\right)+\cos\left(x\right)}{\cos\left(x\right)}}{1+\tan\left(x\right)}$
4

Divide fractions $\frac{\frac{-\sin\left(x\right)+\cos\left(x\right)}{\cos\left(x\right)}}{1+\tan\left(x\right)}$ with Keep, Change, Flip: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$

$\frac{-\sin\left(x\right)+\cos\left(x\right)}{\cos\left(x\right)\left(1+\tan\left(x\right)\right)}$
5

Multiply the single term $\cos\left(x\right)$ by each term

$\frac{-\sin\left(x\right)+\cos\left(x\right)}{\cos\left(x\right)+\tan\left(x\right)\cos\left(x\right)}$

Applying the tangent identity: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$

$\frac{\sin\left(x\right)}{\cos\left(x\right)}\cos\left(x\right)$

Multiplying the fraction by $\cos\left(x\right)$

$\frac{\sin\left(x\right)\cos\left(x\right)}{\cos\left(x\right)}$

Simplify the fraction $\frac{\sin\left(x\right)\cos\left(x\right)}{\cos\left(x\right)}$ by $\cos\left(x\right)$

$\sin\left(x\right)$
6

Applying the trigonometric identity: $\tan\left(\theta\right)\cdot\cos\left(\theta\right)=\sin\left(\theta\right)$

$\frac{-\sin\left(x\right)+\cos\left(x\right)}{\cos\left(x\right)+\sin\left(x\right)}$

Final Answer

$\frac{-\sin\left(x\right)+\cos\left(x\right)}{\cos\left(x\right)+\sin\left(x\right)}$

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